21,701 research outputs found
Low Dimensional Dynamics of the Kuramoto Model with Rational Frequency Distributions
The Kuramoto model is a paradigmatic tool for studying the dynamics of
collective behavior in large ensembles of coupled dynamical systems. Over the
past decade a great deal of progress has been made in analytical descriptions
of the macroscopic dynamics of the Kuramoto mode, facilitated by the discovery
of Ott and Antonsen's dimensionality reduction method. However, the vast
majority of these works relies on a critical assumption where the oscillators'
natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which
allows for a convenient closure of the evolution equations from the
dimensionality reduction. In this paper we investigate the low dimensional
dynamics that emerge from a broader family of natural frequency distributions,
in particular a family of rational distribution functions. We show that, as the
polynomials that characterize the frequency distribution increase in order, the
low dimensional evolution equations become more complicated, but nonetheless
the system dynamics remain simple, displaying a transition from incoherence to
partial synchronization at a critical coupling strength. Using the low
dimensional equations we analytically calculate the critical coupling strength
corresponding to the onset of synchronization and investigate the scaling
properties of the order parameter near the onset of synchronization. These
results agree with calculations from Kuramoto's original self-consistency
framework, but we emphasize that the low dimensional equations approach used
here allows for a true stability analysis categorizing the bifurcations
Characterization of local observables in integrable quantum field theories
Integrable quantum field theories in 1+1 dimensions have recently become
amenable to a rigorous construction, but many questions about the structure of
their local observables remain open. Our goal is to characterize these local
observables in terms of their expansion coefficients in a series expansion by
interacting annihilators and creators, similar to form factors. We establish a
rigorous one-to-one characterization, where locality of an observable is
reflected in analyticity properties of its expansion coefficients; this
includes detailed information about the high-energy behaviour of the observable
and the growth properties of the analytic functions. Our results hold for
generic observables, not only smeared pointlike fields, and the characterizing
conditions depend only on the localization region - we consider wedges and
double cones - and on the permissible high energy behaviour.Comment: minor changes, as to appear in Commun. Math. Phys.; 39 pages, 4
figures, 1 vide
Density of non-residues in Burgess-type intervals and applications
We show that for any fixed \eps>0, there are numbers and with the following property: for every prime and every integer
such that p^{1/(4\sqrt{e})+\eps}\le N\le p, the sequence
contains at least quadratic non-residues modulo . We use this
result to obtain strong upper bounds on the sizes of the least quadratic
non-residues in Beatty and Piatetski--Shapiro sequences.Comment: In the new version we use an idea of Roger Heath-Brown (who is now a
co-author) to simply the proof and improve the main results of the previous
version, 14 page
Investigation of sequence features of hinge-bending regions in proteins with domain movements using kernel logistic regression
Background: Hinge-bending movements in proteins comprising two or more domains form a large class of functional movements. Hinge-bending regions demarcate protein domains and collectively control the domain movement. Consequently, the ability to recognise sequence features of hinge-bending regions and to be able to predict them from sequence alone would benefit various areas of protein research. For example, an understanding of how the sequence features of these regions relate to dynamic properties in multi-domain proteins would aid in the rational design of linkers in therapeutic fusion proteins. Results: The DynDom database of protein domain movements comprises sequences annotated to indicate whether the amino acid residue is located within a hinge-bending region or within an intradomain region. Using statistical methods and Kernel Logistic Regression (KLR) models, this data was used to determine sequence features that favour or disfavour hinge-bending regions. This is a difficult classification problem as the number of negative cases (intradomain residues) is much larger than the number of positive cases (hinge residues). The statistical methods and the KLR models both show that cysteine has the lowest propensity for hinge-bending regions and proline has the highest, even though it is the most rigid amino acid. As hinge-bending regions have been previously shown to occur frequently at the terminal regions of the secondary structures, the propensity for proline at these regions is likely due to its tendency to break secondary structures. The KLR models also indicate that isoleucine may act as a domain-capping residue. We have found that a quadratic KLR model outperforms a linear KLR model and that improvement in performance occurs up to very long window lengths (eighty residues) indicating long-range correlations. Conclusion: In contrast to the only other approach that focused solely on interdomain hinge-bending regions, the method provides a modest and statistically significant improvement over a random classifier. An explanation of the KLR results is that in the prediction of hinge-bending regions a long-range correlation is at play between a small number amino acids that either favour or disfavour hinge-bending regions. The resulting sequence-based prediction tool, HingeSeek, is available to run through a webserver at hingeseek.cmp.uea.ac.uk
A cloudy Vlasov solution
We propose to integrate the Vlasov-Poisson equations giving the evolution of
a dynamical system in phase-space using a continuous set of local basis
functions. In practice, the method decomposes the density in phase-space into
small smooth units having compact support. We call these small units ``clouds''
and choose them to be Gaussians of elliptical support. Fortunately, the
evolution of these clouds in the local potential has an analytical solution,
that can be used to evolve the whole system during a significant fraction of
dynamical time. In the process, the clouds, initially round, change shape and
get elongated. At some point, the system needs to be remapped on round clouds
once again. This remapping can be performed optimally using a small number of
Lucy iterations. The remapped solution can be evolved again with the cloud
method, and the process can be iterated a large number of times without showing
significant diffusion. Our numerical experiments show that it is possible to
follow the 2 dimensional phase space distribution during a large number of
dynamical times with excellent accuracy. The main limitation to this accuracy
is the finite size of the clouds, which results in coarse graining the
structures smaller than the clouds and induces small aliasing effects at these
scales. However, it is shown in this paper that this method is consistent with
an adaptive refinement algorithm which allows one to track the evolution of the
finer structure in phase space. It is also shown that the generalization of the
cloud method to the 4 dimensional and the 6 dimensional phase space is quite
natural.Comment: 46 pages, 25 figures, submitted to MNRA
Accurate and efficient description of protein vibrational dynamics: comparing molecular dynamics and Gaussian models
Current all-atom potential based molecular dynamics (MD) allow the
identification of a protein's functional motions on a wide-range of
time-scales, up to few tens of ns. However, functional large scale motions of
proteins may occur on a time-scale currently not accessible by all-atom
potential based molecular dynamics. To avoid the massive computational effort
required by this approach several simplified schemes have been introduced. One
of the most satisfactory is the Gaussian Network approach based on the energy
expansion in terms of the deviation of the protein backbone from its native
configuration. Here we consider an extension of this model which captures in a
more realistic way the distribution of native interactions due to the
introduction of effective sidechain centroids. Since their location is entirely
determined by the protein backbone, the model is amenable to the same exact and
computationally efficient treatment as previous simpler models. The ability of
the model to describe the correlated motion of protein residues in
thermodynamic equilibrium is established through a series of successful
comparisons with an extensive (14 ns) MD simulation based on the AMBER
potential of HIV-1 protease in complex with a peptide substrate. Thus, the
model presented here emerges as a powerful tool to provide preliminary, fast
yet accurate characterizations of proteins near-native motion.Comment: 14 pages 7 figure
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